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nillless positive integers

Source: Netherlands - Dutch NMO 2023 p1

March 25, 2024
number theory

Problem Statement

A number is called nillless if it is integer and positive and contains no zeros. You can make a positive integer nillless by simply omitting the zeros. We denote this with square brackets, for example [2050]=25[2050] = 25 and [13]=13[13] = 13. When we multiply, add, and subtract we indicate with square brackets when we omit the zeros. For example, [[4⋅5]+7]=[[20]+7]=[2+7]=[9]=9[[4 \cdot 5] + 7] = [[20] + 7] = [2 + 7] = [9] = 9 and [[5+5]+9]=[[10]+9]=[1+9]=[10]=1[[5 + 5] + 9] = [[10] + 9] = [1 + 9] = [10] = 1. The following is known about the two numbers aa and bb: ∙\bullet aa and bb are nillless, ∙\bullet 1<a<b<1001 < a < b < 100, ∙\bullet [[a⋅b]−1]=1[[a \cdot b] - 1] = 1. Which pairs (a,b)(a, b) satisfy these three requirements?
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